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For example, if : [,] is the Dirichlet function that is on irrational numbers and on rational numbers, and [,] is equipped with Lebesgue measure, then the support of is the entire interval [,], but the essential support of is empty, since is equal almost everywhere to the zero function.
The support function is a convex function on . Any non-empty closed convex set A is uniquely determined by h A. Furthermore, the support function, as a function of the set A, is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of ...
The Dirac comb of period 2 π, although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrapped Dirac delta function. It represents a discrete probability distribution concentrated at 2 π n — a degenerate distribution — but the notation treats it as if it were a continuous distribution.
In mathematics, the support (sometimes topological support or spectrum) of a measure on a measurable topological space (, ()) is a precise notion of where in the space the measure "lives". It is defined to be the largest ( closed ) subset of X {\displaystyle X} for which every open neighbourhood of every point of the set has positive measure.
The astroid provides an example of a non-convex hedgehog. [5] The deltoid, an example of a projective hedgehog, showing line segments rotating through the family of lines whose envelope is the deltoid. An example of a projective hedgehog, defined from an anti-symmetric support function, is given by the deltoid curve.
The 1d bump function ().. The function : given by = { (), | | <,, | |, is an example of a bump function in one dimension. It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded closed support.
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Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.